MATH 613 Modular forms (Topics in Number Theory)

Text:

The course will be largely based on James Milne's notes and F. Diamond and J. Shurman "Introduction to modular forms" (available online at UBC library). We will also somtimes use the classic textbook T. Miyake, "Modular forms" (available online at UBC library). We will occasionally refer to several other sources, including:

• William Stein, "Modular Forms, a computational approach"
• S. Lang, "Introduction to modular forms" and "Elliptic Functions".

Classes: Tue, Th 2-3:30pm in CEME (Civil and Mechanical Engineering), Floor 1, room 1210.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: for now, by appointment.

• Course outline

Announcements

---

HOMEWORK

There will be approximately bi-weekly homework assignments.

• Here are some resources if you you need resources for using TeX.
• Problem set 1 (due January 22). Please also read Milne, section on Riemann surfaces as ringed spaces (pp.16-18), including the short section on differential forms.
• Problem set 2 (due February 13). The TeX file of the questions .

  ---

Detailed Course outline

Short descriptions of each lecture and relevant additional references will be posted here as we progress.

• Tuesday January 6 : Lecture 1. Overview and motivation; the definition of modular forms; Riemann surfaces (approximately matching Milne's Introduction).
• Thursday January. 8 : The goal (to be achieved next class) is to define the structure of a Riemann surface on "H mod Gamma". We did: 1. A quick survey of actions of topological groups (see Milne, Section 1 (pp. 13-14) and Part 1 of the homework 1) with the end result that "H mod Gamma" has Hausdorff quotient topology;
  2. Classification of Fractional-linear transformations, with an aside on realizing the Riemann sphere as the projective line over C (see homework). (Milne, pp.30-31, not including Remark 2.11 which will be discussed later).


• Tuesday January 13 : Lecture 3: 1. Realizing H as a quotient of SL_2(R). (Milne, Proposition 2.1 pp.25-26) 2. the fundamental domain for Gamma(1). (Milne, pp.32-33). 3. The complex structure on H mod Gamma(1). (Milne, pp.35-37, up to but not including the genus computation).
• Thursday Jan. 15 : Lecture 4: Cusps and the complex structure on H^* mod Gamma(1). The complex structure on H^* mod Gamma(N).


• Tuesday Jan. 20 : Lecture 5: Review of the preliminaries on complex analysis on Riemann surfaces. Ringed spaces; sheaves, differential forms. (Milne, pp.16-18). Meromorphic functions on Riemann surfaces (Milne, pp.19-20). Stopped at the definition of divisors and principal divisors. On Thursday will continue from there. If you missed the class, please just read Milne pp.16-21 and you'll be right where we stopped.
• Thursday Jan. 22 : Lecture 6: Review of the analysis/geometry, continued: Riemann-Roch theorem and the notion of genus (Milne, pp.18-23).
  See an illuminating mathoverflow discussion of Riemann-Roch,
  as well as Terry Tao's blog .
  Riemann-Hurwitz formula. (Milne, pp.37-39).


• Tuesday January 27: Summary of what we did so far; the genus of the modular surface X(N). Started discussing lattices.
• Thursday January 29: Lecture 9. Weierstrass p-function and the structure of an elliptic curve on C/\Lambda. Definition of Eisenstein series. (Milne, pp.43-47).


• Tuesday February 3 : Lecture 10. Approximately pp.41-43 and 47-50 in Milne.
  If unfamailiar with discriminants, check out this concise and useful note (or any classic algebra textbook).
• Thursday February 5 : NO CLASS!